Problem:
18. Answer these questions for the poset
({{1}, {2}, {4}, {1,2}, {1,4}, {2,4}, {3,4},{1,3,4}, {2,3,4}}, $\subseteq$)
$\quad$a.Find the maximal elements
$\quad$b.Find the minimal elements
$\quad$c.Is there a greatest element
$\quad$d.Is there a least element
$\quad$e.Find all upper bounds of {{2}, {4}}
$\quad$f.Find the least upper bound of {{2}, {4}}, if it exists
$\quad$g.Find the all lower bounds of {{1, 3, 4}, {2, 3, 4}}
$\quad$h.Find the greatest lower bound of {{1, 3, 4}, {2, 3, 4}}, if it exists
My Work
$\quad$a.Maximal elements are {1, 3, 4} and {2, 3, 4}.
$\quad$b.Minimal elements are {1}, {4}, {2}
$\quad$c.There is no greatest element
$\quad$d.There is no least element
$\quad$e.Upper bounds of {{2},{4}} are {{2,4}, {2,3,4}}
$\quad$f.Least upper bound of {{2},{4}} is {2,4}
$\quad$g.Lower bounds of {{1, 3, 4}, {2, 3, 4}} are {{3,4},{1},{4}}
$\quad$h.Greatest lower bound of {{1, 3, 4}, {2, 3, 4}} is {3,4}
Did I miss anything?
In (a) you missed the maximal element $\{1,2\}$. In (g) $\{1\}$ is not a lower bound of $\{1,3,4\}$ and $\{2,3,4\}$: it’s not a subset of the latter. Your answer to (h) is correct, but it’s inconsistent with your (incorrect) answer to (g), since $\{1\}\nsubseteq\{3,4\}$. Everything else is fine.